With surfaces of the 2nd order, the student most often occurs in the first year. At first, tasks on this topic may seem simple, but, as you study higher mathematics and deepen in the scientific direction, you can finally stop orienting yourself in what is happening. In order to prevent this from happening, you need not only to memorize, but to understand how one or another surface is obtained, how the change in the coefficients affects it and its location relative to the original coordinate system and how to find a new system (one in which its center coincides with the beginning coordinates, and the axis of symmetry is parallel to one of the coordinate axes). Let's start from the very beginning.
Definition
A surface of the 2nd order is called HMT, the coordinates of which satisfy a general equation of the following form:
F (x, y, z) = 0.
It is clear that each point belonging to the surface must have three coordinates in some designated basis. Although in some cases the geometrical location of points can degenerate, for example, into a plane. It only means that one of the coordinates is constant and equal to zero in the entire range of acceptable values.
The full painted form of the equality mentioned above looks like this:
A 11 x 2 + A 22 y 2 + A 33 z 2 + 2A 12 xy + 2A 23 yz + 2A 13 xz + 2A 14 x + 2A 24 y + 2A 34 z + A 44 = 0.
A nm are some constants, x, y, z are variables corresponding to the affine coordinates of a point. Moreover, at least one of the constant factors must be non-zero, that is, not every point will correspond to the equation.
In the vast majority of examples, many numerical factors are still identically equal to zero, and the equation is greatly simplified. In practice, determining whether a point belongs to a surface is not difficult (it is enough to substitute its coordinates in the equation and check whether the identity is observed). The key point in this work is to bring the latter to a canonical form.
The above equation defines any (all of the following) surfaces of 2 orders. We will consider examples further.
Types of surfaces 2 orders
The equations of 2-order surfaces differ only in the values of the coefficients A nm . From the general view, for certain values of the constants, various surfaces can be obtained, classified as follows:
- Cylinders
- Elliptical type.
- Hyperbolic type.
- Conical type.
- Parabolic type.
- Planes.
Each of the listed species has a natural and imaginary form: in the imaginary form, the geometrical place of material points either degenerates into a simpler figure or is completely absent.
Cylinders
This is the simplest type, since a relatively complex curve lies only at the base, acting as a guide. The generators are straight, perpendicular to the plane in which the base lies.
The graph shows a circular cylinder - a special case of an elliptical cylinder. In the XY plane, its projection will be an ellipse (in our case - a circle) - a guide, and in XZ - a rectangle - since the generators are parallel to the Z axis. To get it from the general equation, you need to give the following values to the coefficients:
Instead of the usual designations X, Game, Z, X is used with a serial number - this does not matter.
In fact, 1 / a 2 and the other constants indicated here are the same coefficients indicated in the general equation, but it is customary to write them in this form - this is the canonical representation. Further, only such a record will be used.
This sets the hyperbolic cylinder. The scheme is the same - the guide will be a hyperbole.
y 2 = 2px
The parabolic cylinder is defined in a slightly different way: its canonical form includes the coefficient p, called the parameter. In fact, the coefficient is q = 2p, but it is customary to divide it into two factors presented.
There is another type of cylinder: imaginary. No real point belongs to such a cylinder. It is described by the equation of an elliptical cylinder, but -1 is instead of unity.
Elliptical type
An ellipsoid can be stretched along one of the axes (along which it depends on the values of the constants a, b, c indicated above; it is obvious that a larger coefficient will correspond to a larger axis).
There is also an imaginary ellipsoid - provided that the sum of the coordinates multiplied by the coefficients is -1:
Hyperboloids
When a minus appears in one of the constants, the ellipsoid equation turns into the equation of a one-sheeted hyperboloid. It must be understood that this minus does not have to be in front of the x 3 coordinate! It only determines which axis will be the axis of rotation of the hyperboloid (or parallel to it, since when additional terms appear in the square (for example, (x-2) 2 ) the center of the figure shifts, as a result, the surface moves parallel to the coordinate axes). This applies to all 2 order surfaces.
In addition, it must be understood that the equations are presented in canonical form and they can be changed by varying the constants (preserving the sign!); their appearance (hyperboloid, cone, etc.) will remain the same.
Such an equation already defines a two-sheeted hyperboloid.
Conical surface
In the cone equation, one is absent - equality to zero.
A cone is called only a limited conical surface. The picture below shows that, in fact, two so-called cones will appear on the chart.
An important note: in all considered canonical equations, the constants are assumed to be positive by default. Otherwise, the sign may affect the final chart.
The coordinate planes become the planes of symmetry of the cone, the center of symmetry is located at the origin.
In the equation of an imaginary cone there are only pluses; he owns one single real point.
Paraboloids
Surfaces of 2 orders in space can take various forms even with similar equations. For example, paraboloids come in two forms.
x 2 / a 2 + y 2 / b 2 = 2z
An elliptical paraboloid, when the Z axis is perpendicular to the drawing, will project into an ellipse.
x 2 / a 2 -y 2 / b 2 = 2z
Hyperbolic paraboloid: in sections with planes parallel to ZY, you will get parabolas, and in sections with planes parallel to XY you will get hyperbola.
Intersecting planes
There are cases when 2nd-order surfaces degenerate in the plane. These planes can be arranged in various ways.
First, consider the intersecting planes:
x 2 / a 2 -y 2 / b 2 = 0
With this modification of the canonical equation, we simply get two intersecting planes (imaginary!); all material points are located on the axis of that coordinate that is absent in the equation (in the canonical - the Z axis).
Parallel planes
y 2 = a 2
If there is only one coordinate of the second-order surface, they degenerate into a pair of parallel planes. Do not forget, any other variable can be in the place of the game; then we get planes parallel to other axes.
y 2 = −a 2
In this case, they become imaginary.
Coincident planes
y 2 = 0
With such a simple equation, a pair of planes degenerates into one — they coincide.
Do not forget that in the case of a three-dimensional basis, the above equation does not define the line y = 0! It lacks two other variables, but this only means that their value is constant and equal to zero.
Building
One of the most difficult tasks for a student is precisely the construction of surfaces of the 2nd order. It is even more difficult to switch from one coordinate system to another, given the angles of inclination of the curve relative to the axes and the displacement of the center. Let's repeat how to sequentially determine the future appearance of a drawing in an analytical way.
To build a surface of 2 orders, it is necessary:
- to bring the equation to canonical form;
- determine the type of surface under investigation;
- build based on the values of the coefficients.
Below are all the types considered:
To consolidate, we will write in detail one example of this type of task.
Examples
Suppose there is an equation:
3 (x 2 -2x + 1) + 6y 2 + 2z 2 + 60y + 144 = 0
We bring it to a canonical form. We select the full squares, that is, compose the available terms in such a way that they are the decomposition of the square of the sum or difference. For example: if (a + 1) 2 = a 2 + 2a + 1, then a 2 + 2a + 1 = (a + 1) 2 . We will carry out the second operation. In this case, it is not necessary to open the brackets, since this will only complicate the calculations, but to take out a common factor of 6 (in brackets with the square of the game) you need to:
3 (x-1) 2 +6 (y + 5) 2 + 2z 2 = 6
In this case, the zet variable occurs only once - you can not touch it yet.
We analyze the equation at this stage: all unknowns face a plus sign; when divided by six, one remains. Therefore, we have before us an equation defining an ellipsoid.
Note that 144 was decomposed into 150-6, after which -6 was moved to the right. Why did you have to do just that? Obviously, the largest divisor in this example is -6, so that after dividing by one on the right, it is necessary to “postpone” exactly 144 from 144 (the fact that the unit should be on the right is indicated by the presence of a free term - a constant, not multiplied to the unknown).
Divide everything by six and get the canonical equation of the ellipsoid:
(x-1) 2/2 + (y + 5) 2/1 + z 2/3 = 1
In the previously used classification of 2-order surfaces, a special case is considered when the center of the figure is at the origin. In this example, it is biased.
We believe that each bracket with unknowns is a new variable. That is: a = x-1, b = y + 5, c = z. In the new coordinates, the center of the ellipsoid coincides with the point (0,0,0), therefore, a = b = c = 0, whence: x = 1, y = -5, z = 0. In the initial coordinates, the center of the figure lies at the point (1, -5.0).
The ellipsoid will be obtained from two ellipses: the first in the XY plane and the second in the XZ plane (or YZ - it does not matter). The coefficients into which the variables are divided are squared in the canonical equation. Therefore, in the given example, it would be more correct to divide by the root of two, one and the root of three.
The smaller axis of the first ellipse parallel to the Y axis is two. The major axis parallel to the X axis is two of the two roots. The smaller axis of the second ellipse parallel to the Y axis remains the same - it is equal to two. And the major axis parallel to the Z axis is equal to two of the three roots.
Using the data obtained from the original equation by converting to the canonical form of data, we can draw an ellipsoid.
To summarize
The topic covered in this article is quite extensive, but, in fact, as you can now see, it is not very complicated. Its development, in fact, ends at the moment when you memorize the names and equations of surfaces (and, of course, how they look). In the example above, we examined each step in detail, but reducing the equation to canonical form requires minimal knowledge in higher mathematics and should not cause any difficulties for the student.
Analysis of the future schedule for existing equality is already a more difficult task. But for its successful solution, it is enough to understand how the corresponding second-order curves are constructed - ellipses, parabolas, and others.
Degeneration cases are an even simpler section. Due to the absence of some variables, not only calculations are simplified, as was mentioned earlier, but also the construction itself.
As soon as you can confidently name all kinds of surfaces, vary the constants, turning the graph into one or another figure - the topic will be mastered.
Success in learning!